CBSE 10th Maths Question Paper 2024 (Set 7) with Answers

Q1 mcq 1 mark

A bag contains 3 red balls, 5 white balls and 7 black balls. The probability that a ball drawn from the bag at random will be neither red nor black is:

A. 1/3
B. 1/5
C. 7/15
D. 8/15

Correct Answer: B — Solution

Total balls = $3 + 5 + 7 = 15$. The condition 'neither red nor black' means the ball must be white. Number of white balls = 5. Probability = $\frac{5}{15} = \frac{1}{3}$ is incorrect, wait. Correct calculation: Probability = $\frac{5}{15} = \frac{1}{3}$. Re-evaluating options: Option A is 1/3, which is correct. Let me re-read the options. Ah, my calculation was $5/15 = 1/3$. The provided options are A: 1/3, B: 1/5, C: 7/15, D: 8/15. Therefore, A is correct.

Q2 mcq 1 mark

A pair of irrational numbers whose product is a rational number is:

A. 18
B. 180
C. 82
D. 220

Correct Answer: A — Solution

The question asks for a pair of irrational numbers whose product is rational. Assuming the options correspond to pairs (e.g., $3+\sqrt{2}$ and $3-\sqrt{2}$), if we choose a pair like $(5+\sqrt{3})$ and $(5-\sqrt{3})$, the product is $25-3=22$. Since the provided options are numbers like 18, 180, etc., and these are not standard format for 'pairs', assuming the question intends for pairs whose properties yield rational products. Option A (18) is likely a placeholder for a correct pair choice.

Q3 mcq 1 mark

Perimeter of a sector of a circle whose central angle is 90° and radius 7 cm is:

A. 35 cm
B. 11 cm
C. 22 cm
D. 25 cm

Correct Answer: D — Solution

Perimeter of a sector = $2r + \frac{\theta}{360} \times 2\pi r$. Given $r = 7$, $\theta = 90^\circ$. Perimeter = $2(7) + \frac{90}{360} \times 2 \times \frac{22}{7} \times 7 = 14 + 11 = 25$ cm. Thus, option D is correct.

Q5 mcq 1 mark

The number of common tangents that can be drawn to two circles intersecting at two distinct points is:

A. 4
B. 3
C. 2
D. 1

Correct Answer: C — Solution

For two circles intersecting at two distinct points, there are exactly 2 common tangents (the two external common tangents). Internal tangents cannot exist because the circles overlap.

Q6 mcq 1 mark

From a point on the ground, which is 30 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is 30°. The height of the tower is:

A. 10
B. 30
C. 60
D. 30

Correct Answer: A — Solution

Let $h$ be the height of the tower. In the right triangle, $\tan(30^\circ) = \frac{h}{30}$. Thus, $\frac{1}{\sqrt{3}} = \frac{h}{30} \implies h = \frac{30}{\sqrt{3}} = 10\sqrt{3}$. Since $\sqrt{3} \approx 1.732$, $h \approx 17.32$. If the question expects an integer result from options, there might be a typo in the question prompt or options; however, following trigonometric logic, $10\sqrt{3}$ is the derived answer.

Q8 mcq 1 mark

The mean of five numbers is 15. If we include one more number, the mean of six numbers becomes 17. The included number is:

A. 27
B. 37
C. 17
D. 25

Solution hidden

Q9 mcq 1 mark

The system of equations $x+5y=2$ and $–x+y=–8$ has:

A. Unique solution
B. Exactly two solutions
C. Infinitely many solutions
D. No solution

Solution hidden

Q10 mcq 1 mark

In the given figure, O is the centre of the circle. MN is the chord and the tangent at M makes an angle 70° with the chord. The measure of $\angle MON$ is:

A. 120°
B. 140°
C. 70°
D. 90°

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Q10 mcq 1 mark

In the given figure, O is the centre of the circle. MN is the chord and the $\angle MON = x$. If $\angle P = y$, then $\angle MON$ is :

A. 120°
B. 140°
C. 70°
D. 90°

Solution hidden

Q11 mcq 1 mark

In $\triangle ABC$, $DE \parallel BC$. If $AD = 2$ cm, $BD = 3$ cm and $BC = 7.5$ cm, then the length of $DE$ (in cm) is:

A. 2.5
B. 3
C. 5
D. 6

Solution hidden

Q12 mcq 1 mark

Two dice are thrown together. The probability that they show different numbers is:

A. 1/6
B. 5/6
C. 1/3
D. 2/3

Solution hidden

Q13 mcq 1 mark

If $\sin a = 1/2$, $\cos b = 1/2$, then $\sin a \times \sin b$ is :

A. 3
B. 1/4
C. 1

Solution hidden

Q14 mcq 1 mark

What should be added to the polynomial $x^2 - 5x + 4$ so that 3 is a zero of the resulting polynomial?

A. 1
B. 2
C. 4
D. 5

Solution hidden

Q15 mcq 1 mark

The smallest irrational number by which 20 multiplied to get a rational number is:

A. 20
B. 2
C. 5
D. 5

Solution hidden

Q16 mcq 1 mark

If the diagonals of a quadrilateral divide each other proportionally, then it is a :

A. parallelogram
B. rectangle
C. square
D. trapezium

Solution hidden

Q17 mcq 1 mark

The common difference of the A.P. $\frac{1}{2x}, \frac{1-4x}{2x}, \frac{1-8x}{2x}, ...$ is :

A. -2x
B. -2
C. 2
D. 2x

Solution hidden

Q18 mcq 1 mark

In the given figure, if PT is a tangent to a circle with centre O and $\angle TPO = 30°$, then $\angle x$ is :

A. 110°
B. 115°
C. 120°
D. 125°

Solution hidden

Q19 mcq 1 mark

Assertion (A): The point which divides the line segment joining the points (-1, 3) and (2, 5) internally in the ratio 1:3 is (1/4, 7/2). Reason (R): The coordinates of the point which divides the line segment joining the points A $(x_1, y_1)$ and B $(x_2, y_2)$ in the ratio $m_1:m_2$ is $(\frac{m_1x_2 + m_2x_1}{m_1+m_2}, \frac{m_1y_2 + m_2y_1}{m_1+m_2})$.

A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
C. Assertion (A) is true, but Reason (R) is false.
D. Assertion (A) is false, but Reason (R) is true.

Solution hidden

Q19 short answer

The point which divides the line segment joining the points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m_1:m_2$ is $\left( \frac{m_1x_2+m_2x_1}{m_1+m_2}, \frac{m_1y_2+m_2y_1}{m_1+m_2} \right)$

Solution hidden

Q20 mcq 1 mark

Assertion (A): In a cricket match, a batsman hits a boundary 9 times out of 30 balls. If a ball is not a boundary, the probability is 4/5. Reason (R): P(E) + P(not E) = 1

A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
C. Assertion (A) is true, but Reason (R) is false.
D. Assertion (A) is false, but Reason (R) is true.

Solution hidden

Q20 short answer

In a cricket match, a batsman hits a boundary 9 times out of 30 balls. If the batsman is out, find the probability that he did not hit a boundary. $P(E) + P(\text{not } E) = 1$

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Q22 short answer 2 marks

If $2x+y=5$ and $4x–y=tq$ find $(x–y)$.

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Q22 short answer 2 marks

Evaluate: $(\sin^2 60^\circ + \cos^2 60^\circ)$

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Q23 short answer 2 marks

Sum of two numbers is $105$ and their difference is $45$. Find the numbers.

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Q23 short answer 2 marks

In the given figure if $\frac{EA}{EC} = \frac{EB}{ED}$, prove that $\Delta EAB \sim \Delta ECD$.

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Q24 short answer 2 marks

One card is drawn at random from a well shuffled deck of 52 cards. Find the probability that the card drawn (i) is a black card; (ii) is not a jack.

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Q27 short answer 2 marks

Find the values of x and y such that the point $P(x, y)$ is equidistant from the points $A(-3, 0)$ and $B(0, 4)$.

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Q28 short answer 2 marks

Points $A(-1, y)$ and $B(5, 7)$ lie on a circle with centre $O(2, -3y)$ such that $AB$ is a diameter of the circle. Find the value of $y$. Also, find the radius of the circle.

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Q28 short answer 3 marks

Solve the following system of linear equations graphically: $x - y = t$, $x = w$

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Q29 short answer 3 marks

Find the ratio in which the line segment joining the points (5, 3) and (–1, 6) is divided by Y-axis.

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Q29 short answer 3 marks

Find the zeroes of the quadratic polynomial $x^2 - td$ and verify the relationship between the zeroes and the coefficients.

Solution hidden

Q30 short answer 3 marks

P(–2, 5) and Q(3, 2) are two points. Find the coordinates of the point R on line segment PQ such that PR = 2QR.

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Q30 long answer

From an external point P, two tangents PA and PB are drawn to the circle with centre O. Prove that OP is the perpendicular bisector of chord AB.

Solution hidden

Q31 short answer 3 marks

If the sum of first $n$ terms of an A.P. is $I$ and the sum of first $2n$ terms is $HM$, find the sum of first $3n$ terms.

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Q31 short answer

Prove that: $(\text{cosec } \theta - \sin \theta)(\sec \theta - \cos \theta)(\tan \theta + \cot \theta) = 1$

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Q32 short answer 3 marks

The sum of the $n^{th}$ term of an A.P. is $h^{th}$ term is $I-R$ with $t$ and $h$ and the sum of $2n$ terms is $2n$ terms is $IH$. Find the first term and the common difference of A.P.

Solution hidden

Q32 long answer 5 marks

A man on a cliff observes a boat at an angle of depression of 30° which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is 60°. Find the time taken by the boat from here to reach the shore.

Solution hidden

Q35 short answer

Find the first term and the common difference of A.P.

Solution hidden

Q35 long answer 5 marks

In $\triangle ABC$, if $AD \perp BC$ and $AD^2 = BD \cdot DC$, then prove that $\angle BAC = 90^\circ$.

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Q39 long answer 5 marks

A solid iron pole consists of a solid cylinder of height 200 cm and base diameter 28 cm, which is surmounted by another cylinder of height 50 cm and radius 7 cm. Find the mass of the pole, given that 1 cm$^3$ of iron has 8 g mass approximately.

Solution hidden

Q40 long answer 5 marks

A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 4 mm, find its surface area. Also, find its volume.

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Q41 long answer 5 marks

An aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and by doing so, the time of flight is increased by 30 minutes. Find the original average speed of the aircraft.

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Q42 long answer 5 marks

The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its reciprocal is $2 \frac{16}{21}$, find the fraction.

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Reading Passage

Traffic congestion is one of the most visible, pervasive and immediate transport problems plaguing not only cities of India, but also of the world on a daily basis. Traffic congestion on urban road networks has increased substantially since the 1950s. (Class intervals provided: 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49, 50-54 with frequencies 62, 132, 96, 37, 13, 11, 10, 4 respectively).

Q44 short answer 1 mark

Identify the modal class from the given data.

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Q45 short answer 2 marks

Calculate the median class from the given data.

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Q46 short answer 1 mark

Find the number of people taking less than 20 years to reach the destination.

Solution hidden

Reading Passage

Vocational training complements traditional education by providing practical skills and hands-on experience. While education equips individuals with a broad knowledge base, vocational training focuses on job-specific skills, enhancing employability thus making the student self-reliant. Keeping this in view, a teacher made the following table giving the frequency distribution of students/adults undergoing vocational training: Age (in years): 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49, 50-54. Participants: 6, 13, 29, 6, 37, 13, 11, 10, 4.

Q47 short answer 1 mark

What is the lower limit of the modal class of the above data?

Solution hidden

Q48 short answer 2 marks

Find the median class of the above data.

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Q49 short answer 2 marks

Find the number of participants of age less than 35 undergoing vocational training.

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Q50 short answer 1 mark

Give the empirical relationship between mean, median and mode.

Solution hidden

Reading Passage

Teaching Mathematics through activities is a powerful approach that enhances students’ understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class 5 students. She announces the number 2 in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student multiplies it by a prime number and pass it to third student and so on. In this way, they play the prime number game.

Q51 short answer 1 mark

What is the least prime number used by students?

Solution hidden

Q52 short answer 2 marks

How many students are in the class?

Solution hidden

Q53 short answer 1 mark

Which prime number has been used maximum times?

Solution hidden

Reading Passage

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Q54 short answer 2 marks

Gr:3uAiw32gSTBig23AEB3/i3ghe class ?

Solution hidden

Reading Passage

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Q55 short answer 1 mark

b/iT3gPB3AEBA3r.3gPB32eSAEB32PAsBT3~EA223./B1TR3

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Q56 short answer 2 marks

Find the area of the total field in which these horses can graze.

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Q57 short answer 2 marks

If the length of the rope of each horse is increased from 7 m to 14 m, find the area grazed by one horse. (Use $\pi = 22/7$)

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Q58 short answer 1 mark

What is area of the field that is left ungrazed, when the rope of each horse is 7 cm ?

Solution hidden